Tapping Kinetic Energy for Power

Module 3, Lesson 5

Objective: By the end of this lesson you will be able to calculate the power generated by harnessing the kinetic energy of a moving air or water. You’ll see that both wind turbines and water turbines can actually be used to harness a substantial amount of power.

Introduction

The total energy available in wind and the power that can feasibly be extracted from it will be determined using the fundamentals of kinematics. The limitations of location and machinery of wind turbines that restrict the amount of power that can be harnessed will also be examined, before comparing the impact of wind energy with various other common sources of energy.

How is wind energy created?

Wind energy has been proposed as an alternative energy source, although it is currently in an early stage of large scale development [1]. Windmills were seen in Persia as early as 500 AD and were used to grind grain and pump water [2]. It is only in modern times that humanity has attempted to extract raw power from wind in the form of electricity. The question to ask in this early developmental stage is whether or not it is possible to extract a useful amount of raw energy from the wind. To do so, we will consider the broad energetics of wind turbines to determine if harnessing wind energy could be a viable option given constraints of time, location and machinery.

The first thing to consider is whether or not there is enough energy in wind to make it possible to extract a useful amount of power. It is important to note that although wind may possess a lot of kinetic energy, that is the energy resulting from the movement of masses, the rate at which this energy can be extracted limits the amount of useful power available, as power in its most rudimentary form is defined as the rate of doing useful work. To discover whether or not a useful amount of power is available, we must first discuss where the wind comes from and how much power is available in the wind.

Wind energy ultimately comes from a series of energy transformations from solar energy (radiation) to wind energy (kinetic), where about 2% of the solar energy absorbed by the Earth goes into wind [3]. Solar radiation is absorbed by the surface of the Earth and heats it unevenly [4]. Different areas of the globe receive varying amounts of the incident solar intensity (W/m2) due to the angle of the Sun; the equator receives a greater percentage of solar intensity (and hence becomes hotter) than the poles [4]. Also, during the day the land heats up faster than the sea does, while at night the water retains heat longer than the land does [4]. Wind is a direct result of solar heating and the earth's rotation as they generate changes in temperature. As the air gets warmer, it rises and cooler air must rush in to take its place, producing wind! In all, location effects how much wind energy is available (Fig. 1).

Figure 1. A wind energy map of Canada showing the average power (in W/m2 of turbine cross- section area) that can theoretically be extracted from the wind [5].

How much energy can be harnessed by wind?

The mean intercepted solar intensity at the top of the Earth's atmosphere is 350 W/m2. Given that 2% is converted to wind, this results in 7 W/m2 going into wind energy [6]. This wind energy is spread out over the Earth's atmosphere, with 35% of the energy (2.45 W/m2) dissipating in the ﬁrst kilometre above Earth's surface [6]. This energy is dissipated through surface roughness and boundary effects, but could potentially be tapped by wind mills.

Question: Using the information above, calculate how much energy is available in the 1 km boundary layer of the earth to be harnessed over the period of one year. Hint: The surface area of the earth is 5.1x1014 m2

Answer: which is 200 times larger than our energy consumption on Earth, estimated to be 2 x 1020 J [6].

Now we can calculate the energy and power harnessed from the wind. To use the basic equation for kinetic energy, KE = ½mv2, we will need the rate at which air passes through the rotor of the wind turbine. The rotor is made up of the blades that spin on a wind turbine, and the total area of the rotor can be approximated by a circle as this is the area swept by the blades. To do this, we can imagine we are holding a hoop up in the air, and measure the mass of air travelling through the hoop in time Δt (Fig. 2).

Figure 2. At time t = 0, the mass of air is just about to pass through the hoop, but Δt later, the mass of air has passed through the hoop. The mass of this piece of air is the product of its density ρ, area A, and length vΔt.

From this, we can see that the mass is

where ρ is the density of the air (1.2 kg/m3 for standard atmospheric pressure (1 atm) and temperature (0°C) at sea level), v is the velocity of the air and Δt is the length of time for a unit of air to pass through the loop [7]. The area A is the area swept by the blades, not the blade area. This is because the blade moves much faster than the air and so each particle of air is affected by the blade. Substituting the mass of air moving through the blades, which we found above, we ﬁnd the kinetic energy to be

while the power of the wind passing through our hoop is

But this is not the actual power produced by the turbine as turbines can't extract all of the kinetic energy of the wind. Why not? If this was the case the air would stop as soon as it passed through the blades and no other wind would be able to pass through. An analysis by Betz (1919) shows that you cannot capture any more than 59.3% of the wind's energy [8]. To extract his power you want the turbine to slow the wind down by 2/3 of its original speed (as the maximum of P/P0 = 0.593 is found at v2/v1 ≈1/3). A derivation of this can be seen here. Figure ?. A plot of Benz’s Law showing that the peak power output from a turbine occurs when

the velocity of the wind exiting the turbine v2 is 1/3 the velocity of the wind entering the

turbine v1.

The power produced by one turbine is found by

where r is the radius of the circle covered by the rotors (or the length of a blade). The v3 term found here emphasizes the need to have a high wind speed in order to capture an useful amount of power. What we have just derived is based on a single wind turbine in constant wind conditions. In real life, however, wind conditions change. So what local conditions must be satisﬁed in order to make the use of wind turbines feasible?

The location of the wind turbine must be carefully selected. Wind turbines only work efficiently when wind moves uniformly in the same direction. Turbulence, the unsteady flow caused by buildings, trees, and land formations, causes an inconsistent air flow which makes harnessing power inefficient and places increased stress on the rotor. The edge of a continental shelf, high ground and tundra are the best locations to build a turbine [9] because their geography lacks any large obstructions that may create turbulence. Local wind is also an important factor, and should be, on average, at least 7 m/s at 25 m above the Earth's surface in order to make harnessing wind from it worthwhile [3]. One must also keep demand and dependability in mind when considering the extraction of energy from the wind. First of all, since wind is not locally predictable in the short term, the use of wind energy should be limited to only fulfil 5 - 15% of the total energy demand of the area [3]. To overcome this problem and make wind energy more reliable, turbines need to be set up in many different locations so that the power available averages out [3]. In other words, on one day one of three of the locations may not have enough wind to operate, but the next day, that location may be in operation while the other two locations are not. Despite the local wind patterns, however, globally there is always a relatively constant amount of wind energy being harnessed at any one moment.

Question: As a wind turbine technician you monitor the windspeed exiting a turbine to be 1 m/s.

The windspeed around the turbine is 10 m/s. What is the power ratio P/P0 for these wind speeds? Should you call in the maintenance crew?

Answer: From the graph above we see that v2/v1 = 0.1, which makes the power ratio about P/

P0 = 0.54-0.56. The turbine doesn’t seem to be operating at optimal efﬁciency. How do wind turbines work?

The machinery of a wind turbine also has its limitation on how much power can be extracted from wind. To begin, let us cement some of the terminology used in describing the structure of wind turbines, before looking at the actual mechanics of converting wind energy into electricity.

Figure 4. A turbine is composed of a foundation, a tower, a nacelle and a rotor consisting of 3 blades.

Principle components of a wind turbine unit are a foundation, a transformer, a tower, a rotor and a nacelle (Fig. 4) [10]. The wind turns the rotor, which turns the generator to produce electricity. The electricity is then transmitted to a transformer at the base of the tower before going to a substation. To maximize the power extracted, the nacelle, which connects the rotor to the tower and houses the generator, can be rotated into the direction of the wind. Rotors' diameters range from 27 m for a 225 kW generator to 80 m for a 2500 kW generator, and depend on the desired power output, location limitations et cetera (Fig. 5) [8]. A 1 MW turbine has a rotor diameter of 54 m, a tower standing 80m tall, and works in wind speeds ranging from 3 - 25 m/s (10 - 90 km/ h) [7].

Figure 5. The dimensions and characteristics of a typical smaller sized turbine.

The power produced by a wind turbine depends on rotor area, air density, wind speed, and wind shear. Air density increases with colder temperatures, decreased altitude, and decreased humidity. The molar mass of air (29 g/mol) is greater than the molar mass of water (18 g/mol) and so the less moisture in the air, the denser the air is. Wind shear is a difference in wind speed and direction over a short distance and is caused by mountains, coastlines and weather patterns [8]. Wind speed increases the farther you get away from the ground (Fig. 6) [7]. To maximize the power output of wind turbines, rotors are tilted slightly upwards. Why do you think this is? Figure 6. As you get higher off the ground, the air speed increases, corresponding to a longer arrow. The rotors are tilted slightly upwards so that each part of the rotor is exposed to the same speed.

While it may be possible to use a single wind turbine for personal energy demands, entire cities and countries need huge wind farms to satisfy their energy needs. To optimize energy production in a wind farm, turbines are spread 5 - 9 rotor diameters apart in the prevailing wind direction and 3 - 5 rotor diameters apart in the perpendicular direction (Fig. 7) [7]. Figure 7. On a wind farm, turbines must be spaced out enough so that they do not interfere with each other. As the wind passes through the turbine it slows down, and so there is no point in putting a turbine in the region where the air is guaranteed to be slow. One common way of spacing them out is ensuring there is at least 5 rotor diameters between each turbine.

When the turbines are placed on a square grid, the power per unit land area is

where n is the number of turbine diameters between turbines. The average power of a wind turbine farm is the product of the capacity of the farm and the fraction of the time when the wind conditions are near optimal. The capacity factor is usually around 15 - 30% [7].

How does wind energy compare to other energy production alternatives?

Now that it is established that wind is a possible source of power, the beneﬁts and drawbacks need to be considered. Why use wind power in lieu of other energy sources? The harnessing of wind power does not produce hazardous wastes, use non-renewable resources or cause signiﬁcant amounts of damage to the environment [1].

Some CO2 is produced in the manufacturing of the turbines, but as demonstrated in the problem set, it is much less than the emissions from burning an energy-equivalent amount of coal or natural gas. In addition, the use of wind power can reduce hidden costs such as those related to pollution and the consequential healthcare impacts, and in the longer term, climate change [1]. Also, wind turbines use less space than traditional power stations, because you can farm around them, so they can be built without extensively reducing agricultural land and locally- owned wind farms create income for communities [1].

So why, in light of these positive elements, is there so much resistance against wind turbines? Arguments against include the following fears of damages from collapsing turbines, noise, a less attractive skyline, an unreliable power source, unnecessarily high bird fatality, and signiﬁcantly modifying the Earth's wind patterns. First of all, the noise of a typical turbine is 45 dB at 250 m away [11]. This level is lower than the background noise at an ofﬁce or a home [11]. Secondly, the reliability of wind energy increases depending on location and how many farms are operating in a variety of sites within the area. In regards to bird deaths, in the US less than 40,000 are said to die from turbine blades while hundreds of millions are said to die from domestic cats [12]!

Finally, in regards to modifying the Earth's climate, it is plausible that one would see local climate change surrounding areas with a high concentration of wind farms, but the large-scale climatic effects will likely be negligible [13]. In addition, since wind turbines will be replacing coal-ﬁred power plants, if anything, we anticipate a considerable reduction in CO2 emissions. While currently wind turbines are the accepted machinery used to extract wind energy, technology can change quite easily as people discover new and more efﬁcient ways to harness power. Currently, wind cells are being investigated by Accio Energy, which takes a completely different approach to the method of extracting wind energy than the tradition wind turbine [14].

Tidal Power

The water in the ocean is another abundantly flowing fluid that has kinetic energy we can use to generate energy. Specifically, the gravitational pull of the moon causes two tidal bulges, one on the side of the planet closest to earth and one on the opposite side, to move around the Earth. This means the water gets raised and lowered with a period of about 12 hours (if you’re curious why there are two bulges and not just one, check out this page). When this raising and lower occurs in a bay, water must rush in and out through the neck of the bay. We can use the motion of this rushing water to generate energy in a very similar way we use wind turbines. We will see later that there are also ways of harnessing the potential energy of the tidal water. For now we will focus on the kinetic energy. Turbines that use the kinetic energy of the water look almost identical to wind turbines and are governed by exactly the same equations and laws. The technology is still new and the turbines are often referred to as tidal streams.

Figure ?. An artists impression of tidal steam technology taken from wikipedia.

Let’s estimate how much energy is available from an underwater turbine placed in the Burrard Inlet. The first task is to find a place for the turbine to go. Our turbine equation depends on the velocity cubed, so we want it in the place with high water velocity. The continuity equation from fluid dynamics tells us that at two points in a fluid flow the volume of water entering must be the same as the volume leaving. The continuity equation tells us that the fastest moving water will be found at the narrowest part of the channel. Looking at the map of the Burrard Inlet we see this happens at the First Narrows, under the Loin’s Gate bridge, and the Second Narrows, under the appropriately named Second Narrows Bridge.

Figure ?. A map of the Burrard Inlet taken from wikipedia. Where would you place a turbine?

We want to find the velocity of the water in the Narrows. The continuity equation also allows us to equate the volume of water that flows through one of the Narrows with the change in the height of the water level due to the rising and falling tides. Imagine that a volume of water Adx flows under the Second Narrows in a given time dt. As before A cross sectional area of the Narrows and dx is some length of water that moves under the bridge in time dt. The water in the Inlet must rise in time dt to accommodate this influx of water. The volume of the water in the Inlet changes by adh, where a is the surface area of the water in the Inlet and dh it the amount the water rose by in time dt. This is just the continuity equation. We can equate the volumes of water, If this happens in a time dt, we can write the equation as

which gives us the velocity of the water in the Narrows,

To ﬁnd the velocity of water, and thus the power generated by a single turbine all we need to ﬁnd is the rate of change of the height of water in the Inlet, the surface area of the Inlet a, and the cross-sectional area of the Narrows A.

Question: Calculate the maximum velocity of water in the Second Narrows. You can estimate the rate of the change in the height of the Inlet by looking at the Deep Cove tide forecast.

To ﬁnd the fastest speed take the biggest change height over the smallest time interval. The fastest may be the receding tide.

You can ﬁnd the cross-sectional area of the Second Narrows and the surface area of the Inlet past the Second Narrows by looking at Table 1 of this old Coastal Engineering paper.

Answer: A=48,500 ft2. a = 452.7x106 ft2. dh/dt = 8.87ft/ 8h31m = 8.87ft/30660s = 2.89x10-4 ft/s/.

Plugging these in we get 2.70 ft/s or 0.822 m/s or 1.59 knots. Above: Making full use of the wind, the Wolfe Island Wind Farm, Lake Ontario

1. a. b. c. d. Cullum A, Kwan C, Macdonald K. British Columbia Wind Energy Feasibility Study (online).http://www.geog.ubc.ca/courses/geog376/students/class05/cskwan/intro.html [4 May 2009].

2. Dodge, Darrel. Part 1 - Early History Through 1875: Wind Power's Beginnings (online). Illustrated History of Wind Power Development.http://www.telosnet.com/wind/early.html [10 June 2009]. 3. a. b. c. d. Aubrecht GJ. Solar Energy: Wind, Photovoltaics, and Large-Scale Installatons. In: Energy - Physical, Environmental, and Social Impact (3), edited by Erik Fahlgren. Upper Saddle River, NJ: Pearson Education Inc., 2006, chapt. 21, 461-465.

4. a. b. c. Kump, L.R., Kasting, J.F., and Crane, R.G. The Atmospheric Circulation System. In: The Earth System (2), edited by Patrick Lynch. Upper Saddle River, New Jersy, USA: 2004, chapt. 4, pp. 55-82.

5. Environment Canada. Canadian Atlas Level 0 (online). http://collaboration.cmc.ec.gc.ca/ science/rpn/modcom/eole/CanadianAtlas0.html[20 May 2009].

6. a. b. c. Gustavson MR. Limits to Wind Power Utilization. Science 204: 13 - 17, 1979.

7. a. b. c. d. e. MacKay DJC. Sustainable Energy - Without the Hot Air (Online). UIT Cambridge.http://www.inference.phy.cam.ac.uk/sustainable/book/tex/ps/253.326.pdf [4 May 2009].

8. a. b. c. Danish Wind Industry Association. Guided Tour on Wind Energy (online). http:// guidedtour.windpower.org/en/tour/wres/index.htm[4 May 2009].

9. Learning (online). Solacity Inc. http://www.solacity.com/SiteSelection.htm [20 May 2009].

10. D'Emil B, Jacobsen M, Jensen MS, Krohn S, Petersen KC, and Sandstørm, K. Wind with Miller (online). Danish Wind Industry Association. http://windwithmiller.windpower.org/en/kids/ index.htm [4 May 2009].

11. a. b. Clarke S. Electricity Generation Using Small Wind Turbines At Your Home Or Farm (Online). Ontario Ministry of Agriculture, Foods and Rural Affairs. http://www.omafra.gov.on.ca/ english/engineer/facts/03-047.htm#noise [25 May 2009].

12. Marris E and Fairless D. Wind Farms' Deadly Reputation Hard to Shift. Nature 447: 126, 2007.

13. Keith D. Wind Power and Climate Change (online). University of Calgary. http:// www.ucalgary.ca/~keith/WindAndClimateNote.html [20 May 2009]. 14. Accio Energy. About Accio Energy (online). http://accioenergy.com/about.html [12 June 2009].